|
%I #46 Aug 15 2018 12:36:56
%S 1,-744,159768,-36866976,8507424792,-1963211493744,453039686271072,
%T -104545516658693952,24125403112135458840,-5567288717204029449672,
%U 1284733088879405339418768,-296470902355240575283208928,68414985730612787485819011168
%N Coefficients in expansion of E_6/E_4.
%C Also coefficients in expansion of E_10/E_8. - _Seiichi Manyama_, Jun 20 2017
%H Seiichi Manyama, <a href="/A288261/b288261.txt">Table of n, a(n) for n = 0..422</a>
%F From _Seiichi Manyama_, Jun 27 2017: (Start)
%F Let j_0 = 1 and j_1 = j - 744. Define j_m by j_m = j1 | T_0(m), where T_0(m) = mT_{m, 0} is the normalized m-th weight zero Hecke operator. a(n) = j_n((-1+sqrt(3)*i)/2).
%F G.f.: Sum_{n >= 0} j_n((-1+sqrt(3)*i)/2)*q^n. (End)
%F a(n) ~ (-1)^n * 3 * exp(Pi*sqrt(3)*n). - _Vaclav Kotesovec_, Jun 28 2017
%F G.f.: -q*j'/j where j is the elliptic modular invariant (A000521). - _Seiichi Manyama_, Jul 12 2017
%e G.f.: 1 - 744*q + 159768*q^2 - 36866976*q^3 + 8507424792*q^4 - 1963211493744*q^5 + 453039686271072*q^6 + ...
%e From _Seiichi Manyama_, Jun 27 2017: (Start)
%e a(0) = j_0((-1+sqrt(3)*i)/2) = 1,_
%e a(1) = j_1((-1+sqrt(3)*i)/2) = -744 + 0^1 = -744,
%e a(2) = j_2((-1+sqrt(3)*i)/2) = 159768 - 1488*0^1 + 0^2 = 159768. (End)
%t nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5, k]*x^k, {k, 1, nmax}])/(1 + 240*Sum[DivisorSigma[3, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 28 2017 *)
%t terms = 13; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[Ei[6]/Ei[4] + O[x]^terms, x] (* _Jean-François Alcover_, Mar 01 2018 *)
%t a[ n_] := With[{j = Series[1728 KleinInvariantJ[ Log[ Series[q, {q, 0, n + 1}]]/(2 Pi I)], {q, 0, n}]}, SeriesCoefficient[ -q D[j, q] / j, {q, 0, n}]]; (* _Michael Somos_, Aug 15 2018 *)
%Y Cf. A004009 (E_4), A110163, A013973 (E_6).
%Y E_{k+2}/E_k: A288877 (k=2), this sequence (k=4, 8), A288840 (k=6).
%Y Cf. A000521 (j), A035230 (-q*j'), A066395 (1/j), A289141.
%K sign
%O 0,2
%A _Seiichi Manyama_, Jun 17 2017
| 